数据访问和迭代次数：资源约束下最大匹配的对偶原始算法
Access to Data and Number of Iterations: Dual Primal Algorithms for Maximum Matching under Resource Constraints
摘要：
在本文中，我们考虑图形算法的计算模型中的空间使用（随机存取存储，除了只读输入）是次线性的边缘数$m$和访问输入数据的限制。这些问题出现在许多自然环境中，特别是在分析MapReduce或类似的算法时，这些算法使用次线性中央处理来模拟约束并行性。在SPAA 2011中，Lattanzi et al.对于具有$n$节点和$m$边的图，通过mapreduce和中央处理的$O(n^{1+1/p})$空间，使用$O(p)$轮迭代过滤，提供了最大匹配的$O(1)$近似。本文主要研究加权非二部最大匹配问题。对于任何恒定的$p>1$，我们提供了一个迭代采样为基础的算法计算$(1-ε)$ -近似的加权非二分最大匹配，使用$O(p/ε)$轮的采样，和$O(n^{1+1/p})$空间。结果扩展到$b$ -匹配，具有微小变化。本文结合自适应素描文献和快速原始-对偶算法的基础上放松Dantzig-Wolfe决策程序。每一轮采样都是通过线性草图实现的，并在MapReduce的单轮中执行。本文还证明了问题的非标准线性松弛，特别是基于惩罚的公式，有助于在mapreduce和类似的设置中减少迭代的自适应依赖性。
英文摘要：
In this paper we consider graph algorithms in models of computation where the space usage (random accessible storage, in addition to the read only input) is sublinear in the number of edges $m$ and the access to input data is constrained. These questions arises in many natural settings, and in particular in the analysis of MapReduce or similar algorithms that model constrained parallelism with sublinear central processing. In SPAA 2011, Lattanzi etal. provided a $O(1)$ approximation of maximum matching using $O(p)$ rounds of iterative filtering via mapreduce and $O(n^{1+1/p})$ space of central processing for a graph with $n$ nodes and $m$ edges. We focus on weighted nonbipartite maximum matching in this paper. For any constant $p>1$, we provide an iterative sampling based algorithm for computing a $(1-ε)$-approximation of the weighted nonbipartite maximum matching that uses $O(p/ε)$ rounds of sampling, and $O(n^{1+1/p})$ space. The results extends to $b$-Matching with small changes. This paper combines adaptive sketching literature and fast primal-dual algorithms based on relaxed Dantzig-Wolfe decision procedures. Each round of sampling is implemented through linear sketches and executed in a single round of MapReduce. The paper also proves that nonstandard linear relaxations of a problem, in particular penalty based formulations, are helpful in mapreduce and similar settings in reducing the adaptive dependence of the iterations.
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